Advanced Molecular Dynamics Velocity scaling Andersen Thermostat Hamiltonian & Lagrangian Appendix A Nose-Hoover thermostat
Naïve approach Velocity scaling Do we sample the canonical ensemble?
Partition function Maxwell-Boltzmann velocity distribution
Fluctuations in the momentum: Fluctuations in the temperature
Andersen thermostat Every particle has a fixed probability to collide with the Andersen demon After collision the particle is give a new velocity The probabilities to collide are uncorrelated (Poisson distribution)
Velocity Verlet:
Andersen thermostat: static properties
Andersen thermostat: dynamic properties
Hamiltonian & Lagrangian The equations of motion give the path that starts at t1 at position x(t1) and end at t2 at position x(t2) for which the action (S) is the minimum S<S t x t2 t1 S<S
Example: free particle Consider a particle in vacuum: v(t)=vav Always > 0!! η(t)=0 for all t
Calculus of variation At the boundaries: η(t) is small η(t1)=0 and η(t2)=0 η(t) is small True path for which S is minimum η(t) should be such the δS is minimal
A description which is independent of the coordinates This term should be zero for all η(t) so […] η(t) Integration by parts If this term 0, S has a minimum Zero because of the boundaries η(t1)=0 and η(t2)=0 Newton A description which is independent of the coordinates
The true path plus deviation Lagrangian Cartesian coordinates (Newton) → Generalized coordinates (?) Lagrangian Action The true path plus deviation
Desired format […] η(t) Partial integration Should be 0 for all paths Equations of motion Conjugate momentum Lagrangian equations of motion
Newton? Valid in any coordinate system: Cartesian Conjugate momentum
Pendulum Equations of motion in terms of l and θ Conjugate momentum
With these variables we can do statistical thermodynamics Lagrangian dynamics We have: 2nd order differential equation Two 1st order differential equations With these variables we can do statistical thermodynamics Change dependence:
Legrendre transformation Example: thermodynamics We have a function that depends on and we would like We prefer to control T: S→T Legendre transformation Helmholtz free energy
Hamilton’s equations of motion Hamiltonian Hamilton’s equations of motion
Newton? Conjugate momentum Hamiltonian
Extended system 3N+1 variables Nosé thermostat Lagrangian Hamiltonian Extended system 3N+1 variables Associated mass Conjugate momentum
Nosé and thermodynamics Delta functions Recall MD MC Gaussian integral Constant plays no role in thermodynamics
Equations of Motion Lagrangian Hamiltonian Conjugate momenta
Nosé Hoover